A birth, death and migration process

1969 ◽  
Vol 6 (3) ◽  
pp. 687-691 ◽  
Author(s):  
S. R. Adke
Keyword(s):  
Time Dependent ◽  
Death Process ◽  
Migration Process ◽  
Death Rates ◽  
And Migration ◽  
Birth Death

A model proposed by Bailey (1968) for migratory individuals which reproduce according to a simple birth-death process is generalized to include time dependent birth and death rates.

A birth, death and migration process

1969 ◽  
Vol 6 (03) ◽  
pp. 687-691 ◽  
Author(s):  
S. R. Adke
Keyword(s):  
Time Dependent ◽  
Death Process ◽  
Migration Process ◽  
Death Rates ◽  
And Migration ◽  
Birth Death

A model proposed by Bailey (1968) for migratory individuals which reproduce according to a simple birth-death process is generalized to include time dependent birth and death rates.

scholarly journals Families of birth-death processes with similar time-dependent behaviour

2000 ◽  
Vol 37 (03) ◽  
pp. 835-849 ◽  
Author(s):  
R. B. Lenin ◽  
P. R. Parthasarathy ◽  
W. R. W. Scheinhardt ◽  
E. A. van Doorn
Keyword(s):  
Death Rate ◽  
General Setting ◽  
Time Dependent ◽  
Death Process ◽  
Similar Time ◽  
Death Rates ◽  
Transition Functions ◽  
Time Dependent Behaviour ◽  
Dependent Behaviour ◽  
Birth Death

We consider birth-death processes taking values in but allow the death rate in state 0 to be positive, so that escape from is possible. Two such processes with transition functions are said to be similar if, for all there are constants c ij such that for all t ≥ 0. We determine conditions on the birth and death rates of a birth-death process for the process to be a member of a family of similar processes, and we identify the members of such a family. These issues are also resolved in the more general setting in which the two processes are called similar if there are constants c ij and ν such that for all t ≥ 0.

A TIME-DEPENDENT STUDY OF THE KNOCKOUT QUEUE

2013 ◽  
Vol 27 (3) ◽  
pp. 309-317
Author(s):  
Brian Fralix
Keyword(s):  
Queue Length ◽  
Queueing Systems ◽  
Time Dependent ◽  
Death Process ◽  
Birth Rates ◽  
Death Rates ◽  
Current State ◽  
Dependent Behavior ◽  
Number Of Customers ◽  
Birth Death

We examine the time-dependent behavior of a birth–death process, whose birth rates and death rates are decreasing and increasing, respectively, with respect to the current state. Such models can be used to describe Markovian queueing systems with exponential reneging, where potential arrivals balk with a certain probability that depends on the number of customers observed upon arrival. Our results are derived by interpreting the birth–death process as the queue-length process of what we refer to as the “knockout queue.”

scholarly journals Families of birth-death processes with similar time-dependent behaviour

2000 ◽  
Vol 37 (3) ◽  
pp. 835-849 ◽  
Author(s):  
R. B. Lenin ◽  
P. R. Parthasarathy ◽  
W. R. W. Scheinhardt ◽  
E. A. van Doorn
Keyword(s):  
Death Rate ◽  
General Setting ◽  
Time Dependent ◽  
Death Process ◽  
Similar Time ◽  
Death Rates ◽  
Transition Functions ◽  
Time Dependent Behaviour ◽  
Dependent Behaviour ◽  
Birth Death

We consider birth-death processes taking values in but allow the death rate in state 0 to be positive, so that escape from is possible. Two such processes with transition functions are said to be similar if, for all there are constants cij such that for all t ≥ 0. We determine conditions on the birth and death rates of a birth-death process for the process to be a member of a family of similar processes, and we identify the members of such a family. These issues are also resolved in the more general setting in which the two processes are called similar if there are constants cij and ν such that for all t ≥ 0.

scholarly journals A Numerical Approach for Evaluating the Time-Dependent Distribution of a Quasi Birth-Death Process

2021 ◽  
Author(s):  
Michel Mandjes ◽  
Birgit Sollie
Keyword(s):  
Continuous Time ◽  
Real Life ◽  
Numerical Approach ◽  
Time Dependent ◽  
Death Process ◽  
Continuous Time Markov Chain ◽  
Likelihood Estimator ◽  
Real Life Data ◽  
The Matrix ◽  
Birth Death

AbstractThis paper considers a continuous-time quasi birth-death (qbd) process, which informally can be seen as a birth-death process of which the parameters are modulated by an external continuous-time Markov chain. The aim is to numerically approximate the time-dependent distribution of the resulting bivariate Markov process in an accurate and efficient way. An approach based on the Erlangization principle is proposed and formally justified. Its performance is investigated and compared with two existing approaches: one based on numerical evaluation of the matrix exponential underlying the qbd process, and one based on the uniformization technique. It is shown that in many settings the approach based on Erlangization is faster than the other approaches, while still being highly accurate. In the last part of the paper, we demonstrate the use of the developed technique in the context of the evaluation of the likelihood pertaining to a time series, which can then be optimized over its parameters to obtain the maximum likelihood estimator. More specifically, through a series of examples with simulated and real-life data, we show how it can be deployed in model selection problems that involve the choice between a qbd and its non-modulated counterpart.

The Relaxation time for truncated birth-death processes

1987 ◽  
Vol 1 (4) ◽  
pp. 367-381 ◽  
Author(s):  
Julian Keilson ◽  
Ravi Ramaswamy
Keyword(s):  
Relaxation Time ◽  
Exit Time ◽  
Relaxation Times ◽  
Dual Process ◽  
Death Process ◽  
Exit Times ◽  
Initial State ◽  
Death Rates ◽  
Absorbing States ◽  
Birth Death

The relaxation time for an ergodic Markov process is a measure of the time until ergodicity is reached from its initial state. In this paper the relaxation time for an ergodic truncated birth-death process is studied. It is shown that the relaxation time for such a process on states {0,1, …, N} is the quasi-stationary exit time from the set {,2, …, N{0,1,…, N, N + 1} with two-sided absorption at states 0 and N + 1. The existence of such a dual process has been observed by Siegmund [15] for stochastically monotone Markov processes on the real line. Exit times for birth- death processes with two absorbing states are studied and an efficient algorithm for the numerical evaluation of mean exit times is presented. Simple analytical lower bounds for the relaxation times are obtained. These bounds are numerically accessible. Finally, the sensitivity of the relaxation time to variations in birth and death rates is studied.

scholarly journals Locally adaptive Bayesian birth-death model successfully detects slow and rapid rate shifts

2019 ◽  
Author(s):  
Andrew F. Magee ◽  
Sebastian Höhna ◽  
Tetyana I. Vasylyeva ◽  
Adam D. Leaché ◽  
Vladimir N. Minin
Keyword(s):  
Random Field ◽  
Markov Random Field ◽  
Death Process ◽  
Rate Variation ◽  
Rapid Rate ◽  
Birth Rates ◽  
Death Rates ◽  
Growth Of Groups ◽  
Markov Random ◽  
Birth Death

AbstractBirth-death processes have given biologists a model-based framework to answer questions about changes in the birth and death rates of lineages in a phylogenetic tree. Therefore birth-death models are central to macroevolutionary as well as phylodynamic analyses. Early approaches to studying temporal variation in birth and death rates using birth-death models faced difficulties due to the restrictive choices of birth and death rate curves through time. Sufficiently flexible time-varying birth-death models are still lacking. We use a piecewise-constant birth-death model, combined with both Gaussian Markov random field (GMRF) and horseshoe Markov random field (HSMRF) prior distributions, to approximate arbitrary changes in birth rate through time. We implement these models in the widely used statistical phylogenetic software platform RevBayes, allowing us to jointly estimate birth-death process parameters, phylogeny, and nuisance parameters in a Bayesian framework. We test both GMRF-based and HSMRF-based models on a variety of simulated diversification scenarios, and then apply them to both a macroevolutionary and an epidemiological dataset. We find that both models are capable of inferring variable birth rates and correctly rejecting variable models in favor of effectively constant models. In general the HSMRF-based model has higher precision than its GMRF counterpart, with little to no loss of accuracy. Applied to a macroevolutionary dataset of the Australian gecko family Pygopodidae (where birth rates are interpretable as speciation rates), the GMRF-based model detects a slow decrease whereas the HSMRF-based model detects a rapid speciation-rate decrease in the last 12 million years. Applied to an infectious disease phylodynamic dataset of sequences from HIV subtype A in Russia and Ukraine (where birth rates are interpretable as the rate of accumulation of new infections), our models detect a strongly elevated rate of infection in the 1990s.Author summaryBoth the growth of groups of species and the spread of infectious diseases through populations can be modeled as birth-death processes. Birth events correspond either to speciation or infection, and death events to extinction or becoming noninfectious. The rates of birth and death may vary over time, and by examining this variation researchers can pinpoint important events in the history of life on Earth or in the course of an outbreak. Time-calibrated phylogenies track the relationships between a set of species (or infections) and the times of all speciation (or infection) events, and can thus be used to infer birth and death rates. We develop two phylogenetic birth-death models with the goal of discerning signal of rate variation from noise due to the stochastic nature of birth-death models. Using a variety of simulated datasets, we show that one of these models can accurately infer slow and rapid rate shifts without sacrificing precision. Using real data, we demonstrate that our new methodology can be used for simultaneous inference of phylogeny and rates through time.

Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process

1985 ◽  
Vol 17 (3) ◽  
pp. 514-530 ◽  
Author(s):  
Erik A. Van Doorn
Keyword(s):  
Complete Solution ◽  
Sufficient Conditions ◽  
Rational Functions ◽  
Difficult Problem ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Decay Parameter ◽  
Death Rates ◽  
Exponential Ergodicity ◽  
Birth Death

This paper is concerned with two problems in connection with exponential ergodicity for birth-death processes on a semi-infinite lattice of integers. The first is to determine from the birth and death rates whether exponential ergodicity prevails. We give some necessary and some sufficient conditions which suffice to settle the question for most processes encountered in practice. In particular, a complete solution is obtained for processes where, from some finite state n onwards, the birth and death rates are rational functions of n. The second, more difficult, problem is to evaluate the decay parameter of an exponentially ergodic birth-death process. Our contribution to the solution of this problem consists of a number of upper and lower bounds.

Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process

1985 ◽  
Vol 17 (03) ◽  
pp. 514-530 ◽  
Author(s):  
Erik A. Van Doorn
Keyword(s):  
Complete Solution ◽  
Sufficient Conditions ◽  
Rational Functions ◽  
Difficult Problem ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Decay Parameter ◽  
Death Rates ◽  
Exponential Ergodicity ◽  
Birth Death

This paper is concerned with two problems in connection with exponential ergodicity for birth-death processes on a semi-infinite lattice of integers. The first is to determine from the birth and death rates whether exponential ergodicity prevails. We give some necessary and some sufficient conditions which suffice to settle the question for most processes encountered in practice. In particular, a complete solution is obtained for processes where, from some finite state n onwards, the birth and death rates are rational functions of n. The second, more difficult, problem is to evaluate the decay parameter of an exponentially ergodic birth-death process. Our contribution to the solution of this problem consists of a number of upper and lower bounds.

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